Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(331\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.109561.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.331.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 6.2.109561.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 2x^{3} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 10\cdot 13 + 2\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 4\cdot 13^{6} + 2\cdot 13^{7} + 6\cdot 13^{8} +O(13^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 2\cdot 13 + 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{5} + 11\cdot 13^{6} + 12\cdot 13^{7} + 5\cdot 13^{8} +O(13^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 2 + \left(3 a + 9\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(10 a + 8\right)\cdot 13^{3} + \left(10 a + 8\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} + \left(9 a + 8\right)\cdot 13^{6} + \left(8 a + 9\right)\cdot 13^{7} + \left(10 a + 5\right)\cdot 13^{8} +O(13^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 3 + \left(9 a + 11\right)\cdot 13 + \left(a + 3\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(2 a + 8\right)\cdot 13^{4} + \left(6 a + 11\right)\cdot 13^{5} + \left(3 a + 10\right)\cdot 13^{6} + \left(4 a + 8\right)\cdot 13^{7} + \left(2 a + 7\right)\cdot 13^{8} +O(13^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 3 + \left(3 a + 5\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(8 a + 1\right)\cdot 13^{4} + \left(10 a + 7\right)\cdot 13^{5} + \left(12 a + 7\right)\cdot 13^{6} + \left(6 a + 5\right)\cdot 13^{7} + \left(4 a + 1\right)\cdot 13^{8} +O(13^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 11 + 9 a\cdot 13 + \left(8 a + 5\right)\cdot 13^{2} + \left(3 a + 2\right)\cdot 13^{3} + \left(4 a + 1\right)\cdot 13^{4} + \left(2 a + 9\right)\cdot 13^{5} + 9\cdot 13^{6} + \left(6 a + 12\right)\cdot 13^{7} + \left(8 a + 11\right)\cdot 13^{8} +O(13^{9})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $6$ | $2$ | $(3,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,6,4)(2,3,5)$ | $0$ |
| $6$ | $4$ | $(1,3,2,6)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.